Based on the previous results, we can now also discuss the emptiness problem, and the closely related finiteness problem. This is particularly interesting since emptiness for a language expressed by a formula \(f\)  and variable \(x\)  corresponds exactly to the satisfiability problem for \(f\) .
Based on existing literature [1]}, [2]}, [3]}, [4]}, it is not hard to show that emptiness and finiteness are decidable for \(\operatorname{VPL}\)  and \(\operatorname{WE}\operatorname{+}\operatorname{REG}\)  but undecidable for \(\operatorname{DCF}\) .
